The Donaldson geometric flow is a local smooth semiflow

TL;DR

This paper proves the local existence and uniqueness of smooth solutions for the Donaldson geometric flow on symplectic forms in four-manifolds, establishing it as a semiflow in a specific functional space.

Contribution

It demonstrates the local well-posedness of the Donaldson geometric flow as a smooth semiflow in Besov spaces for symplectic forms on four-manifolds.

Findings

Existence of unique smooth solutions for the flow

Flow is a semiflow on a Besov space

Applicable to symplectic forms in four dimensions

Abstract

We prove the local existence of unique smooth solutions of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. It is a semiflow on the Besov space $B^{1,p}_2(M, \Lambda^2)$ for $p > 4$. The Donaldson geometric flow was introduced by Simon Donaldson in [2]. For a detailed exposition see [6].